Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{k^2 - 3k - 54}{k^2 + 6k}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{k^2 - 3k - 54}{k^2 + 6k} = \dfrac{(k - 9)(k + 6)}{(k)(k + 6)} $ Notice that the term $(k + 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(k + 6)$ gives: $y = \dfrac{k - 9}{k}$ Since we divided by $(k + 6)$, $k \neq -6$. $y = \dfrac{k - 9}{k}; \space k \neq -6$